Superconducting behavior of a new metal iridate compound, SrIr₂, under pressure

Figure 1(b) shows the temperature (T) dependence of magnetic susceptibility (M/H) of SrIr₂ in zero field cooling (ZFC) and field cooling (FC) modes at 0 GPa. The values of T_c for SrIr₂ was determined to be 6.6 K from the M/H  −  T plot at ZFC mode. The shielding fraction of SrIr₂ was 100% at 2.0 K, indicating that SrIr₂ is a bulk superconductor. A significant diamagnetic transition was observed even in the M/H  −  T plot in FC mode. The result also supports bulk superconductivity.

The XRD patterns of SrIr₂ at 0 GPa are shown in figures 1(c) and (d). As shown in figure 1(c), the XRD pattern of SrIr₂ was well analyzed by LeBail fitting with the space group of Fd$\overline{3}$m (No. 227, choice 2) [19], and is called the 'MgCu₂ type structure'. The lattice constant a was determined to be 7.7968(1) Å. The value of a for SrIr₂ is the same as those (7.545 Å for CaIr₂ and 7.70 Å for SrIr₂) [19] reported previously for MIr₂, indicating that the SrIr₂ is exactly formed. The XRD pattern of SrIr₂ was analyzed by Rietveld refinement using the same space group, as shown in figure 1(d). Table 1 lists the atomic coordinates of SrIr₂; Sr and Ir atoms occupy the special sites of 3/8,3/8,3/8 and 0,0,0, respectively. In the Rietveld refinement, the Debye Waller factor, U_iso, for the atoms were fixed as 0.0125 Ų, and the lattice constant, a, was refined. The a for SrIr₂ was 7.7826(1) Å, and is the same as that (a  =  7.7968(1) Å) determined for SrIr₂ by LeBail fitting. The crystal structure of SrIr₂ is shown in figure S1 in supplementary material (stacks.iop.org/JPhysCM/32/025704/mmedia).

Table 1. Atomic coordinate of SrIr₂. The crystal structure is face-centered cubic ($F{\rm d}\overline{3}{\rm m}$; No. 227).

Atom	Site	Occupancy	x	y	z	B (Å2)
Ir	16c	1.0	0	0	0	0.9870
Sr	8b	1.0	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{3}{8}$	0.9870

The R R(10 K)  −  T plots of two different SrIr₂ samples (samples A and B) are shown in figures 2(a) and (b), respectively. Both samples A and B of SrIr₂ indicated the same T_c (=6.6 K) and 100% shielding fraction at 2 K at ambient pressure. A clear superconducting transition was observed for SrIr₂ in a pressure range of 0 to 17.4 GPa; zero R was unambiguously confirmed. The T_c values of SrIr₂ at 0.78 and 17.4 GPa were determined to be 5.80 and 4.72 K, respectively, from the R/R(10 K)  −  T plots shown in figure 2(b); the T_c is defined as the temperature that is half the value of R in the normal state, as shown in figure S2(a). A zero R was clearly observed below T_c, and the decrease in R is sharp in all pressure range, suggesting a homogenous superconductivity. A metallic behavior in the normal state above T_c is observed for SrIr₂ in all pressure range, as shown in figure 2(c). Specifically, a broad convex shape in the R/R(280 K)  −  T plot is observed, indicating a weak localization of carriers [20, 21]. Here, we briefly discuss the residual R ratio, RRR (=R(room temperature)/R(minimum temperature)). The RRR value of sample A of SrIr₂ was 2.2 at 7.94 GPa, indicating the presence of defects such as impurities and grain boundaries. The values of R(280 K) and R(7 K) of sample A at 7.94 GPa were taken as R(room temperature) and R(minimum), respectively. The RRR value of 2.2 implies small mean free path in normal state, small coherence length ξ, and type-II superconductivity. In addition, such a small RRR for normal metallic state indicates effectively strong electronic scattering that would be explained by defects or disorder (structural, magnetic, etc...), leading to a short mean free path. In the case of superconductors, the above nature will lead to 'dirty limit' features, which means a small coherence length and type II superconductivity.

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The plots of onset T_c ($T_{{\rm c}}^{{\rm onset}}$), T_c, and the temperature indicating zero R ($T_{{\rm c}}^{{\rm 0}}$) for sample A of SrIr₂ are plotted against pressure (figure 3(a)). The T_c  −  p  and $T_{{\rm c}}^{{\rm onset}}$  −  p  plots become almost constant above 7–8 GPa. In fact, T_c appears to exhibit a slight upward turn above ~7 GPa, although this is still unclear because of the limited data. However, if this is true, the upward turn is an interesting behavior because it is unexpected of a BCS-type superconductor.

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Moreover, to confirm the reliability of the plots shown in figure 3(a), the plots of $T_{{\rm c}}^{{\rm onset}}$, T_c, and $T_{{\rm c}}^{{\rm 0}}$ for sample B of SrIr₂ are plotted against pressure (figure 3(b)). From the plots, $T_{{\rm c}}^{{\rm onset}}$ unambiguously exhibited an upward turn above 7–8 GPa; however, the upward turn was not clearly confirmed for the plots of T_c  −  p  and $T_{{\rm c}}^{{\rm 0}}$  −  p . The T_c  −  p  plots for both samples A and B of SrIr₂ shown in figure 3(c) have been found to be similar. However, the T_c for sample A is lower by ~0.25 K compared to that for sample B for the entire pressure range, indicating a degree of experimental error of T_c in this study. The T_c value becomes almost constant above ~7 GPa for both samples A and B (figure 3(c)), after a rapid decrease in a low pressure range. To summarize, the upward turn of T_c or $T_{{\rm c}}^{{\rm onset}}$ against pressure appears to start at approximately 7–10 GPa for both samples of SrIr₂.

To check if the increase in $T_{{\rm c}}^{{\rm onset}}$ above ~7 GPa (figure 3(b)) originates from intrinsic nature of SrIr₂, the superconducting transition width, ΔT_c (=T(90%)  −  T(10%)), for the sample B, where T(90%) and T(10%) refer to the temperature providing 90% and 10% of resistance at normal state just above superconducting transition. The plot is shown in figure 3(d), showing the increase in ΔT_c above 7 GPa. This behavior is somewhat similar to the increase in $T_{{\rm c}}^{{\rm onset}}$, but as described in experimental section, the hydrostaticity should be still maintained even above 7 GPa because NaCl employed as pressure medium guarantees the quasi-hydrostatic pressure up to 25–30 GPa [17, 18].

In addition, the variation of $T_{{\rm c}}^{{\rm onset}}$ above ~7 GPa is larger than that of ΔT_c. As seen from figure 3(b), the T_c and $T_{{\rm c}}^{{\rm 0}}$ became almost constant above ~7 GPa. These results seem to suggest the intrinsic change of electronic structure of SrIr₂, but the possibility of apparent variation of $T_{{\rm c}}^{{\rm onset}}$ caused by non-hydrostatic pressure (inhomogeneity of pressure) also needs to be considered at the present stage. If the observed phenomenon does not originate from the inhomogeneity of pressure, but it reflects a physical feature characteristic of SrIr₂, the mechanism of increasing T_c by applying pressure must be fully discussed, which will be described later. Finally, we may point out the possibility that the $T_{{\rm c}}^{{\rm onset}}$ may be sensitive to the change of electronic structure caused by pressure, in comparison with the T_c and $T_{{\rm c}}^{{\rm 0}}$, because the $T_{{\rm c}}^{{\rm onset}}$ should reflect even filamentous superconductivity.

Figure 4(a) shows the R  −  T plots of sample A of SrIr₂ under an H of 0–5.0 T recorded at 7.94 GPa, which indicates that the decrease in R is clearly suppressed with increasing H, thereby assuring that it can be assigned to a superconducting transition. The plot of H_c2  −  T_c at 7.94 GPa (figure 4(b)) shows a linear relationship; H_c2 at 0 K (H_c2(0)) was evaluated to be 6.7 T and was still lower than the Pauli limit $H_{{\rm p}}^{{\rm BCS}}$ (=1.86T_c), i.e. $H_{{\rm p}}^{{\rm BCS}}$  =  8.7 T. In this H_c2  −  T_c plot, the temperature providing 50% of normal-state resistance at each H is employed as the T_c. The values of the reduced critical field, ${{h}^{*}}\left( T \right)={}^{\left[ \frac{{{H}_{{\rm c2}}}(T)}{{{T}_{{\rm c}}}} \right]}\diagup{}_{{{\left[ -\frac{{\rm d}{{H}_{{\rm c}2}}(T)}{{\rm d}T} \right]}_{T={{T}_{{\rm c}}}}}} =~\frac{{{H}_{{\rm c}2}}(T)}{{{T}_{{\rm c}}}\times {{\left[ -\frac{{\rm d}{{H}_{{\rm c}2}}(T)}{{\rm d}T} \right]}_{T={{T}_{{\rm c}}}}}}=\frac{{{H}_{{\rm c}2}}(T)}{{{H}_{{\rm c}2}}(0)}$, of SrIr₂ at 7.94 GPa were plotted as a function of t ($=$$\frac{T}{{{T}_{{\rm c}}}}$, which are shown in figure 4(c); the H_c2(0) of SrIr₂ at 7.94 GPa is larger than that, 4 T, of CaIr₂ at ambient pressure [13].

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The h^*(t)  −  t plot (figure 4(c)) for sample A of SrIr₂ at 7.94 GPa was drawn based on the data shown in figure 4(a). We attempted to fit the plot using three different models, i.e. the Werthamer–Helfand–Hohenberg (WHH) theory that corresponds to the s-wave dirty limit superconductivity [22, 23]. The WHH theory predicts h^*(0)  =  0.69 at t  =  0, but the experimental h^*(0) does not reach the value. The experimental h^*(t)  −  t plot was well followed by the p -wave polar model (see figure 4(c)). Moreover, the theoretical curve of the s-wave clean limit was not the same as the experimental h^*  −  t plot. The h^*(0) value should typically be in the range of 0.80–0.85 for the p -wave polar model [24–28].

To confirm the validity of the above h^*(t)  −  t plot, we investigated the H dependence of R  −  T plots of sample B of SrIr₂ at 11.3 GPa (figure S2(b) in supplementary material). The H_c2  −  T_c plot is shown in figure S2(c). The H_c2(0) value was evaluated to be 10.0 T from the linear relationship of the H_c2  −  T_c plot. The h^*  −  t plot (figure 4(d)) is well fitted by the p -wave polar model, similar to that for sample A of SrIr₂ at 7.94 GPa, supporting the deviation from the simple s-wave pairing.

In the previous section, we suggested the dirty limit feature from the small RRR for normal metallic state. In fact, the p -wave superconducting pairing or the deviation from s-wave pairing was indicated from the h^* – t plots (figures 4(c) and (d)). These results may imply that SrIr₂ has both natures of the dirty limit feature and deviation from the s-wave. Thus, the pairing mechanism of SrIr₂ may not be simple, although a previous study suggested that superconducting CaIr₂ was categorized as a weak coupled BCS-type superconductor [13]. The other experiments such as Knight shift and angle resolved photoemission spectroscopy (ARPES) may be required for the final determination of superconducting pairing symmetry.

Figure 5(a) shows the XRD patterns of SrIr₂ under pressure (0–22.4 GPa) at 297 K; it indicates that all patterns can be analyzed with the space group of Fd$\overline{3}$m, similarly as the XRD pattern at 0 GPa (figure 1). No structural phase transitions are observed at 0–22.4 GPa. The XRD patterns were analyzed by Rietveld refinement, and these patterns at 9.56 and 22.4 GPa are shown in figures 5(b) and (c), respectively, together with the patterns fitted by Rietveld refinement. In the Rietveld analysis for XRD patterns under pressure, the a was refined and the U_iso was fixed. As shown in figure 6(a), the a value decreases with increasing pressure. All XRD patterns indicated the presence of Ir metal at less than 10%. The fraction (10% of Ir and 90% of SrIr₂) did not change with applied pressure, and the presence of unknown impurities is suggested; peaks due to unknown impurities are indicated by an asterisk (figure 5(a)). As a matter of fact, Rietveld refinement was achieved considering the presence of two phases (main phase of SrIr₂ and minor phase of Ir). Here, it is noteworthy that this sample is different from that employed for the XRD measurement at 0 GPa (figures 1(c) and (d)). The SrIr₂ sample employed for pressure-dependent XRD measurement is categorized as 'sample C', which exhibited 100% shielding fraction at 2 K, with the T_c value being as high as 5.8 K, as confirmed from the M/H  −  T plot (not shown).

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Moreover, we must comment on the hydrostaticity of sample at each pressure. The daphne 7373 was used as pressure medium for XRD measurement, which guarantees the good hydrostatic pressure up to 2.4 GPa [16]. Therefore, the broadened peaks may be observed in the XRD patterns above 2.4 GPa. Actually, the XRD peaks broaden with increasing pressure. The Rietveld refinement for the XRD pattern was successfully performed at 0–22.4 GPa, and the estimated standard deviation (esd) of lattice constant, a, determined was too small, implying the negligible pressure-inhomogeneity in discussing the crystal structure, i.e. the reliable lattice parameters were obtained from the XRD patterns at 0–22.4 GPa.

The p   −  V plot (figure 6(b)) was well fitted using the Birch–Murnaghan formula [29]; the values of V at 0 GPa (V₀), bulk modulus (K₀) at 0 GPa, and derivative of K₀ (K') are evaluated using least-squares fitting. No anomaly was observed in the a  −  p  and p   −  V plots (figures 6(a) and (b), respectively). The a and V values decrease monotonously even at ~8 GPa, in which T_c indicated an upward turn with pressure. The plots of $T_{{\rm c}}^{{\rm onset}}$  −  V, T_c  −  V and $T_{{\rm c}}^{{\rm 0}}$  −  V are provided in figure 6(c), showing that the upward turn of $T_{{\rm c}}^{{\rm onset}}$ occurs at ~440 ų (7–8 GPa), in which any significant variation of V are not observed. This implies that the suggested anomaly in the T_c  −  p  plot of SrIr₂ does not originate from structural transition.

Recently, the upward turn of T_c against pressure was recorded for metal-doped FeSe ((NH₃)_yM_xFeSe) prepared by the liquid NH₃ method [30–32]. In this case, it was found that the electron density exhibited a sudden increase above the pressure, thus leading to an upward turn [31], as observed from the Hall effect measurement. This result was associated with the variation in the Fermi surface (Lifshitz transition). We have not yet performed Hall effect measurements under pressure because of the polycrystalline powder sample. Therefore, it was still unclear if the suggested T_c enhancement was associated with the variation in the electronic structure (variation in the density of states on the Fermi level). This is the future task.